A279510 -id:A279510 - cp's OEIS frontend

A279513 Multiplicative with a(p^k) = p*a(k) for any prime p and k>0.

1, 2, 3, 4, 5, 6, 7, 6, 6, 10, 11, 12, 13, 14, 15, 8, 17, 12, 19, 20, 21, 22, 23, 18, 10, 26, 9, 28, 29, 30, 31, 10, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 44, 30, 46, 47, 24, 14, 20, 51, 52, 53, 18, 55, 42, 57, 58, 59, 60, 61, 62, 42, 12, 65, 66, 67, 68

Offset: 1

Rémy Sigrist, Dec 13 2016

To compute a(n): multiply (with multiplicity) each prime number appearing in the prime tower factorization of n (see A182318 for definition).
a(n) = n if n is squarefree.
a(n) <= A000026(n) for any n>0.
First differs from A000026 at n=256: a(256)=12 and A000026(256)=16.
If n = p_1 * p_2 * ... * p_k with p_1, p_2, ..., p_k primes, then a(p_1 ^ p_2 ^ ... ^ p_k) = n.

			a(6!) = a(2^(2^2)*3^2*5) = 2*2*2*3*2*5 = 240.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.33 Hall-Montgomery Constant, p. 207.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Richard Alan Gilman and Rainer Tschiersch, Problem 6660, The American Mathematical Monthly, Vol. 100, No. 3 (1993), pp. 296-298.

Cf. A000026, A182318, A279510.

a:= proc(n) option remember; `if`(n=1, 1,
      mul(i[1]*a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 22 2020

a[n_] := a[n] = If[n==1, 1, Times @@ (#[[1]] a[#[[2]]]& /@ FactorInteger[n] )]; Array[a, 256] (* Jean-François Alcover, Mar 31 2017 *)

a(n) =  my (f=factor(n)); return (prod(i=1, #f~, f[i,1]*a(f[i,2])))

Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/p^2 + (p-1)*Sum_{k>=2} a(k)/p^(2*k)) = 0.8351076361... (Gilman and Tschiersch, 1993; Finch, 2003; the constant was calculated by Kevin Ford). - Amiram Eldar, Nov 04 2022

A300957 Fixed points of A300955.

Original entry on oeis.org

1, 5, 6, 7, 11, 13, 17, 19, 23, 29, 30, 31, 35, 37, 41, 42, 43, 47, 53, 55, 59, 61, 65, 66, 67, 71, 72, 73, 77, 78, 79, 83, 85, 89, 91, 95, 97, 101, 102, 103, 107, 108, 109, 113, 114, 115, 119, 127, 131, 133, 137, 138, 139, 143, 145, 149, 151, 155, 157, 161

Offset: 1

Rémy Sigrist, Mar 17 2018

nonn

For any n > 0, A279510(A279510(n)) belongs to this sequence (and this sequence is infinite).
For any n > 0:
- a(n) is a multiple of 2 iff a(n) is a multiple of 3,
- if a(n) is a multiple of 2 then A007814(a(n)) = A300955(A007949(a(n))) and A300955(A007814(a(n))) = A007949(a(n)),
- if a prime p > 3 divides a(n), then the p-adic valuation of a(n) belongs to this sequence.
Squarefree numbers coprime to 6 are in this sequence, and all members of this sequence are 0, 1, or 5 mod 6, so the lower density is at least 3/Pi^2 = 0.303... and the upper density is at most 1/2. This could be improved with more care. - Charles R Greathouse IV, May 17 2024

			A300955(42) = 42 hence 42 belongs to this sequence.

Cf. A007814, A007949, A279510, A300955.

b:= n-> `if`(n=1, 1, mul(`if`(i[1]=2, 3, `if`(i[1]=3,
             2, i[1]))^b(i[2]), i=ifactors(n)[2])):
select(n-> n=b(n), [$1..200])[]; # Alois P. Heinz, Mar 17 2018

cp's OEIS Frontend

A279513 Multiplicative with a(p^k) = p*a(k) for any prime p and k>0.

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